The Impact of Aging, Dementia, and SES on Cognitive Decline

A Longitudinal Study Using Linear Mixed Models

Adasia M., Bess T., Preethi R.

Introduction

Linear Mixed Models (LMMs) are powerful statistical tools designed to analyze data with complex structures, such as hierarchical data (e.g., individuals within groups) or repeated measures (e.g., assessments taken over time). Unlike traditional methods, LMMs account for variations at both the group level and the individual level, making them ideal for studying patterns of cognitive decline across aging populations.

Why LMMs?

  • Fixed Effects: Capture overall population trends, like how age or socioeconomic status influences cognitive function on average.
  • Random Effects: Model differences between individuals, accounting for unique trajectories over time or variability across participants.
  • Data Flexibility: Handle missing data and unbalanced datasets effectively, ensuring reliable results even when some observations are incomplete.

Literature Review

Linear Mixed Models: An Overview

  • Extend simple linear regression by incorporating fixed effects (population-level) and random effects (subject-level variability).
  • Ideal for analyzing longitudinal and hierarchical datasets.
  • Provide unbiased estimates even with missing data (Bates, 2014; Gelman & Hill, 2007).

Covariance Structure

  • Essential for modeling dependencies between repeated measures (Starkweather, 2010).
  • Positive covariance: variables move together.
  • Negative covariance: variables move in opposite directions.
  • Allows explicit modeling of within-subject variability, critical for longitudinal studies.

Literature Review Continued

Robust Estimation

  • Methods that minimize the influence of outliers, ensuring parameter reliability (Agostinelli & Yohai, 2016).
  • Non-robust techniques like OLS regression are highly sensitive to extreme values and may yield biased results.

Challenges of Traditional Methods

  • Traditional linear models assume independence of observations, often violated in clustered data (Barr et al., 2013).

  • Handling missing data through listwise deletion or imputation can introduce bias (Enders, 2010).

Methods

Our Focus

This project applies LMMs to investigate:

  • The influence of age, dementia status, and socioeconomic factors on cognitive decline.

  • Cognitive function measured using the Mini-Mental State Examination (MMSE), a widely-used tool for assessing cognitive impairment.

  • How repeated assessments over time help us understand individual changes in cognitive function while capturing broader population trends.

Methods

Dataset: OASIS-Longitudinal MRI Data in Nondemented and Demented Older Adults

Summary of the chosen dataset:

  • Includes MRI scans of 150 subjects aged 60-96.
  • Longitudinal data collected over 373 sessions.
  • Subjects scanned at least twice, with visits separated by at least a year.
  • Clinical Dementia Rating (CDR) categorized as nondemented or with mild Alzheimer’s disease.
  • 72 were nondemented throughout, while 64 were demented initially and remained so.
  • Participants were right-handed, consisting of 62 men and 88 women.
  • Each session had 3-4 T1-weighted MRI scans.

Methods

Modeling Approach

  • Outcome Variable: Mini-Mental State Exam (MMSE) score.

  • Fixed Effects: Age, dementia status, SES, and interactions.

  • Random Effects: Subject ID to capture individual variability.

  • Software: R (packages: lme4, tableone for data summaries).

R script for Table One

#install.packages("tableone")
library(tableone)

#install.packages("readr")
library(readr)

oasis_longitudinal <- read_csv("C:/IDC6940/IDC6940_BDP/oasis_longitudinal.csv")

# Load necessary library
library(tableone)

# Recode M/F for easier handling in Table 1 (optional)
oasis_longitudinal$Gender <- ifelse(oasis_longitudinal$'M/F' == "M", "Male", "Female")

# Define the variables to be included in Table 1
vars <- c("Age", "Gender", "Hand", "EDUC", "SES", "MMSE", "CDR", "eTIV", "nWBV", "ASF")

# Define the stratifying variable (Group)
strata <- "Group"

# Create Table One
table1 <- CreateTableOne(vars = vars, strata = strata, data = oasis_longitudinal, factorVars = c("Gender", "Hand", "CDR"))

# Print the table with p-values
print(table1, showAllLevels = TRUE, smd = TRUE)
                  Stratified by Group
                   level  Converted        Demented         Nondemented     
  n                            37              146              190         
  Age (mean (SD))           79.76 (7.43)     76.26 (6.94)     77.06 (8.10)  
  Gender (%)       Female      24 ( 64.9)       60 ( 41.1)      129 ( 67.9) 
                   Male        13 ( 35.1)       86 ( 58.9)       61 ( 32.1) 
  Hand (%)         R           37 (100.0)      146 (100.0)      190 (100.0) 
  EDUC (mean (SD))          15.46 (2.52)     13.67 (2.90)     15.14 (2.74)  
  SES (mean (SD))            1.73 (0.96)      2.77 (1.20)      2.39 (1.05)  
  MMSE (mean (SD))          28.68 (1.56)     24.51 (4.50)     29.23 (0.88)  
  CDR (%)          0           18 ( 48.6)        0 (  0.0)      188 ( 98.9) 
                   0.5         19 ( 51.4)      102 ( 69.9)        2 (  1.1) 
                   1            0 (  0.0)       41 ( 28.1)        0 (  0.0) 
                   2            0 (  0.0)        3 (  2.1)        0 (  0.0) 
  eTIV (mean (SD))        1459.27 (135.43) 1485.85 (173.77) 1495.50 (184.89)
  nWBV (mean (SD))           0.72 (0.04)      0.72 (0.03)      0.74 (0.04)  
  ASF (mean (SD))            1.21 (0.11)      1.20 (0.14)      1.19 (0.14)  
                  Stratified by Group
                   p      test SMD   
  n                                  
  Age (mean (SD))   0.045       0.313
  Gender (%)       <0.001       0.371
                                     
  Hand (%)             NA      <0.001
  EDUC (mean (SD)) <0.001       0.433
  SES (mean (SD))  <0.001       0.652
  MMSE (mean (SD)) <0.001       1.041
  CDR (%)          <0.001       5.599
                                     
                                     
                                     
  eTIV (mean (SD))  0.510       0.149
  nWBV (mean (SD)) <0.001       0.465
  ASF (mean (SD))   0.683       0.111

Table One

level Converted Demented Nondemented p
n 37 146 190
Female F      24 ( 64.9)      60 ( 41.1)     129 ( 67.9) <0.001
Male M      13 ( 35.1)      86 ( 58.9)      61 ( 32.1)
Dom Hand R      37 (100.0)     146 (100.0)     190 (100.0)     NA
CDR 0      18 ( 48.6)       0 (  0.0)     188 ( 98.9) <0.001
0.5      19 ( 51.4)     102 ( 69.9)       2 (  1.1)
1       0 (  0.0)      41 ( 28.1)       0 (  0.0)
2       0 (  0.0)       3 (  2.1)       0 (  0.0)
Age   79.76 (7.43)   76.26 (6.94)   77.06 (8.10) 0.045
Education   15.46 (2.52)   13.67 (2.90)   15.14 (2.74) <0.001
SES    1.73 (0.96)    2.77 (1.20)    2.39 (1.05) <0.001
MMSE   28.68 (1.56)   24.51 (4.50)   29.23 (0.88) <0.001
eTIV 1459.27 (135.43) 1485.85 (173.77) 1495.50 (184.89) 0.51
nWBV    0.72 (0.04)    0.72 (0.03)    0.74 (0.04) <0.001
ASF    1.21 (0.11)    1.20 (0.14)    1.19 (0.14) 0.683

Data Visualizations

Age of Participants

Data Visualizations

Gender of Participants

Data Visualizations

Socioeconomic Status of Participants

Data Visualizations

Normalized Whole Brain Volume vs Age

The Role of Age in Predicting Cognitive Decline: An Analysis Using MMSE Scores

By: Adasia McClinton

Data Cleaning and Ingestion

data <- read.csv("C:/IDC6940/IDC6940_BDP/oasis_longitudinal.csv")
library(mice)
imputed_data <- mice(data, m = 5, method = 'pmm')

 iter imp variable
  1   1  SES  MMSE
  1   2  SES  MMSE
  1   3  SES  MMSE
  1   4  SES  MMSE
  1   5  SES  MMSE
  2   1  SES  MMSE
  2   2  SES  MMSE
  2   3  SES  MMSE
  2   4  SES  MMSE
  2   5  SES  MMSE
  3   1  SES  MMSE
  3   2  SES  MMSE
  3   3  SES  MMSE
  3   4  SES  MMSE
  3   5  SES  MMSE
  4   1  SES  MMSE
  4   2  SES  MMSE
  4   3  SES  MMSE
  4   4  SES  MMSE
  4   5  SES  MMSE
  5   1  SES  MMSE
  5   2  SES  MMSE
  5   3  SES  MMSE
  5   4  SES  MMSE
  5   5  SES  MMSE
imputed_list <- lapply(1:5, function(i) complete(imputed_data, i))

Model 0: Age as the Sole Predictor

  • Assess whether age alone is a significant predictor of cognitive decline as measured by MMSE scores.

\text{MMSE}_{ij} = 29.660 - 0.031 \times \text{Age}_{ij} + u_{0j} + \epsilon_{ij}

  • Intercept: 29.660 (baseline MMSE score).
  • Age Coefficient: -0.031 (small, non-significant decline with age).

Key Findings

  • Age alone is not a significant predictor of MMSE scores.
  • Indicates additional variables may explain cognitive decline.

Model 0: Age as the Sole Predictor

R Code

# Model 0: Age as the sole predictor
library("lme4")
library("mitml")
library("Matrix")
model0 <- lapply(imputed_list, function(data) lmer(MMSE ~ Age + (1 | Subject.ID), data = data))
pooled_results <- testEstimates(model0, method = "D2")
summary(pooled_results)

Call:

testEstimates(model = model0, method = "D2")

Final parameter estimates and inferences obtained from 5 imputed data sets.

             Estimate Std.Error   t.value        df   P(>|t|)       RIV       FMI 
(Intercept)    29.782     2.457    12.120 82281.404     0.000     0.007     0.007 
Age            -0.034     0.032    -1.061 76583.901     0.289     0.007     0.007 

Unadjusted hypothesis test as appropriate in larger samples.
  • Results: The age effect on MMSE was small and non-significant, suggesting that age alone does not fully explain cognitive decline.

Model 1: Adding Dementia Severity and SES

  • Examine the combined effect of age, dementia severity (CDR), and SES

    \text{MMSE}_{ij} = 34.544 - 0.070 \times \text{Age}_{ij} - 4.699 \times \text{GD}_{ij} - 0.231 \times \text{SES}_{ij} + 0.008 \times (\text{Age} \times \text{GD})_{ij} + u_{0j} + \epsilon_{ij}

  • Age Coefficient: -0.070 (non-significant decline).

  • CDR Coefficient: -4.699 (significant decline in MMSE).

  • SES Coefficient: -0.231 (small decline).

Key Findings

  • Dementia severity significantly predicts MMSE scores.

  • Age and SES are not significant predictors in this model.

Model 1: Adding Dementia Severity and SES

R Code

model1 <- lapply(imputed_list, function(data) {
  lmer(MMSE ~ Age * Group + SES + (1 | Subject.ID), data = data)
})
pooled_results <- testEstimates(model1, method = "D2")
summary(pooled_results)

Call:

testEstimates(model = model1, method = "D2")

Final parameter estimates and inferences obtained from 5 imputed data sets.

                      Estimate Std.Error   t.value        df   P(>|t|)       RIV       FMI 
(Intercept)             34.702     6.468     5.366 4.054e+05     0.000     0.003     0.003 
Age                     -0.070     0.081    -0.867 1.876e+06     0.386     0.001     0.001 
GroupDemented           -4.438     7.276    -0.610 9.751e+05     0.542     0.002     0.002 
GroupNondemented        -4.232     6.946    -0.609 3.402e+06     0.542     0.001     0.001 
SES                     -0.309     0.202    -1.527 2.444e+02     0.128     0.147     0.135 
Age:GroupDemented        0.007     0.092     0.075 6.199e+05     0.940     0.003     0.003 
Age:GroupNondemented     0.063     0.088     0.719 1.932e+06     0.472     0.001     0.001 

Unadjusted hypothesis test as appropriate in larger samples.
  • Results: This model showed no significant effects for age, SES, or age-by-group interactions, indicating that additional factors may better explain cognitive decline.

Final Model: Significant Predictors Only

Focus on significant predictors: CDR and nWBV.

\text{MMSE}_{ij} = 17.894 - 5.441 \times \text{CDR}_{ij} + 15.038 \times \text{nWBV}_{ij} + u_{0j} + \epsilon_{ij}

  • CDR Coefficient: -5.441 (strong negative effect).

  • nWBV Coefficient: 15.038 (positive effect).

Key Findings

  • CDR strongly predicts cognitive decline.

  • Higher nWBV mitigates decline.

Final Model: Significant Predictors Only

R Code

final_model <- lapply(imputed_list, function(data) {
  lmer(MMSE ~ CDR + nWBV + (1 | Subject.ID), data = data)
})
pooled_final_results <- testEstimates(final_model, method = "D2")
summary(pooled_final_results)

Call:

testEstimates(model = final_model, method = "D2")

Final parameter estimates and inferences obtained from 5 imputed data sets.

             Estimate Std.Error   t.value        df   P(>|t|)       RIV       FMI 
(Intercept)    17.743     3.628     4.891 34385.802     0.000     0.011     0.011 
CDR            -5.275     0.462   -11.425  2668.419     0.000     0.040     0.039 
nWBV           15.193     4.898     3.102 39475.575     0.002     0.010     0.010 

Unadjusted hypothesis test as appropriate in larger samples.
  • Results: CDR was significantly associated with MMSE scores, showing that higher dementia severity leads to lower cognitive function. Additionally, higher nWBV was associated with higher MMSE, supporting its role in cognitive preservation.

Conclusions

Key Takeaways

  • Age alone does not predict cognitive decline.

  • Dementia severity (CDR) is a strong negative predictor.

  • Higher brain volumes (nWBV) are protective against cognitive decline.

Brain Volume: The Role of Age and Socioeconomic Status

By: Preethi R.

Fitting the Linear Mixed Model:

  • Outcome: nWBV (Normalized Whole Brain Volume) – representing structural brain changes over time.

  • Primary Predictor: Age to see if aging correlates with changes in brain volume.

  • Secondary Predictor: Socioeconomic Status (SES) to observe if socioeconomic factors impact brain volume over time.

  • Fixed Effects: Age and Socioeconomic Status (SES).

  • Random Effects: Subject.ID.

The linear mixed model used in this analysis is given by:

\text{nWBV}{ij} = \beta_0 + \beta_1 \cdot \text{Age}{ij} + \beta_2 \cdot \text{SES}{ij} + \mu_{i} + \epsilon_{ij}

Where:

  • nWBV_{ij} : Normalized whole brain volume for subject (i) at time (j).

  • \beta_0 : Overall intercept (fixed effect).

  • \beta_1 : Fixed effect of Age.

  • \beta_2 : Fixed effect of SES.

  • \mu_{i} : Random intercept for each subject.

  • \epsilon_{ij} : Residual error term.

METHODOLOGY


The most common method used in fitting linear mixed models are:


1. Maximum Likelihood Estimation (MLE):

  • it determines the parameters under which the observed data is most probable.

  • estimates both the fixed effects (population-level parameters) and variance components (random effects and residual variances) by maximizing the likelihood of the observed data.


2. Restricted Maximum Likelihood Estimation (REML) :

  • maximizes the likelihood of the data after adjusting for the fixed effects, focusing on variance components estimation.

  • less biased because it adjusts for the loss of degrees of freedom caused by estimating fixed effects.

Data Cleaning:


#install.packages("readr")
library(readr)

oasis_data <- read_csv("oasis_longitudinal.csv")

# Remove duplicate rows
oasis_data <- oasis_data[!duplicated(oasis_data), ]

# Remove rows with missing values
oasis_data <- na.omit(oasis_data)

# View cleaned data
head(oasis_data)

1. Age as the only predictor


#install.packages("lme4")
library(lme4)
oasis_data$gender <- oasis_data$'M/F'
oasis_data$SubjectID <- oasis_data$'Subject ID'

# Fit the linear mixed model with only Age as the predictor, reml
model <- lmer(nWBV ~ Age + (1 | SubjectID), data = oasis_data)
summary(model)
Linear mixed model fit by REML ['lmerMod']
Formula: nWBV ~ Age + (1 | SubjectID)
   Data: oasis_data

REML criterion at convergence: -1852.4

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-4.0245 -0.4668  0.0150  0.4481  3.6450 

Random effects:
 Groups    Name        Variance  Std.Dev.
 SubjectID (Intercept) 1.009e-03 0.031772
 Residual              6.971e-05 0.008349
Number of obs: 354, groups:  SubjectID, 142

Fixed effects:
              Estimate Std. Error t value
(Intercept)  0.9977560  0.0171965   58.02
Age         -0.0034837  0.0002208  -15.78

Correlation of Fixed Effects:
    (Intr)
Age -0.988

equation and interpretation of age as the only predictor


the linear mixed equation is: \text{nWBV}{ij} = 0.9978 - 0.0035 \cdot \text{Age}{ij} + \mu_{i} + \epsilon_{ij}

where:

• 0.9978 is the fixed intercept, representing the average baseline nWBV when Age is 0.

• -0.0035 is the fixed effect estimate for Age, indicating that for each additional year of age, the nWBV decreases by approximately 0.0035 units on average.


INTERPRETATION:

  • age has a significant negative effect on nWBV

  • t-value for Age = -15.78

  • this shows a strong association between age and decreasing nWBV.

2.Age and Socio-Economic status as predictors


  • lmer function uses REML unless specified otherwise.
# Fit Linear Mixed Model with Age and Socio Economic status as the predictors
model <- lmer(nWBV ~ Age + SES + (1 | SubjectID), data = oasis_data)
summary(model)
Linear mixed model fit by REML ['lmerMod']
Formula: nWBV ~ Age + SES + (1 | SubjectID)
   Data: oasis_data

REML criterion at convergence: -1842.3

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-4.0215 -0.4692  0.0140  0.4432  3.6478 

Random effects:
 Groups    Name        Variance  Std.Dev.
 SubjectID (Intercept) 0.0010160 0.031874
 Residual              0.0000697 0.008349
Number of obs: 354, groups:  SubjectID, 142

Fixed effects:
              Estimate Std. Error t value
(Intercept)  0.9956305  0.0183229  54.338
Age         -0.0034846  0.0002211 -15.761
SES          0.0008823  0.0024137   0.366

Correlation of Fixed Effects:
    (Intr) Age   
Age -0.933       
SES -0.342  0.015

equation and interpreation of age and ses:


The model equation: \text{nWBV}{ij} = 0.9956 - 0.0035 \cdot \text{Age}{ij} + 0.0009 \cdot \text{SES}{ij} + \mu_{i} + \epsilon_{ij} where:

• 0.9956 is the intercept, representing the estimated nWBV when both Age and SES are 0.

• -0.0035 is the coefficient for Age, indicating that each additional year of age is associated with an average decrease in nWBV by approximately 0.0035 units.

• 0.0009 is the coefficient for SES, suggesting that for each unit increase in SES, there is a slight positive association with nWBV, though it is not statistically significant (t-value = 0.366).


INTERPRETATION

  • Age

    • strong, statistically significant negative effect on nWBV

    • consistent with prior findings (t-value of -15.761)

  • SES

    • small positive effect on nWBV as t-value = 0.366

    • SES may not contribute meaningfully to explaination of variation in nWBV in this model.

Maximum Likelihood (predictors: age and ses)


R script for MLE :

# Fit the model with ML for model comparison
model_ml <- lmer(nWBV ~ Age + SES + (1 | SubjectID), data = oasis_data, REML = FALSE)
summary(model_ml)
Linear mixed model fit by maximum likelihood  ['lmerMod']
Formula: nWBV ~ Age + SES + (1 | SubjectID)
   Data: oasis_data

     AIC      BIC   logLik deviance df.resid 
 -1867.5  -1848.2    938.8  -1877.5      349 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-4.0300 -0.4722  0.0137  0.4444  3.6503 

Random effects:
 Groups    Name        Variance  Std.Dev.
 SubjectID (Intercept) 9.979e-04 0.031589
 Residual              6.953e-05 0.008338
Number of obs: 354, groups:  SubjectID, 142

Fixed effects:
              Estimate Std. Error t value
(Intercept)  0.9952308  0.0182348  54.579
Age         -0.0034795  0.0002202 -15.804
SES          0.0008834  0.0023926   0.369

Correlation of Fixed Effects:
    (Intr) Age   
Age -0.933       
SES -0.341  0.015

Interpretation for Maximum Likelihood:

  • Random Effects:

    • (Intercept) Variance: 0.0009979 (SD: 0.0316) This is the variability in nWBV across different subjects.

    • Residual Variance: 0.00006953 (SD: 0.00834) This is the remaining variance in nWBV after accounting for both fixed and random effects.


  • Fixed Effects:

    • Intercept: 0.9952. likely represents an estimated baseline close to 1.

    • Age: -0.00348. nWBV decreases with increasing age.

      • for each one-unit increase in Age, nWBV is expected to decrease by about 0.00348 units, holding SES constant.
    • t value: -15.804 (a high magnitude t-value) - Age is statistically significant

    • SES: 0.00088. This positive coefficient suggests that as SES increases, there’s a very slight increase in nWBV.

      • low t-value (0.369), SES may not have a statistically significant effect on nWBV in this model.


INTERPRETATION

Age is a significant predictor of nWBV,

  • with a negative effect indicating cognitive decline as age increases.

SES appears to have little to no significant effect on nWBV based on this model.

Comparison of the two methods:


Fixed Effects: The estimates for Age and SES under REML are nearly identical to those under ML, showing consistent results.

Age has a significant negative impact on nWBV, indicating cognitive decline with age.

SES appears to have little to no significant effect on nWBV.



In summary,

  • REML is appropriate for final model interpretation as it provides the better estimates for variance components,

  • while both methods confirm Age as a key predictor of nWBV decline.

Modeling CDR: Dementia and Education

By: Bess T.

Fitting the Linear Mixed Model:

  • Outcome: CDR (Clinical Dementia Rating) – Examining dementia severity over time.

  • Primary Predictor: Dementia Status (Demented vs. Nondemented) to explore how dementia impacts CDR progression.

  • Secondary Predictor: Education Level (EDUC) to assess if education affects dementia severity.

  • Fixed Effects: Dementia Status and Education Level (EDUC).

  • Random Effects: Subject.ID.

The linear mixed model used in this analysis is given by:

\text{CDR}{ij} = \beta_0 + \beta_1 \cdot \text{DementiaStatus}{ij} + \beta_2 \cdot \text{EDUC}{ij} + \mu_{i} + \epsilon_{ij}

Where:

  • CDR_{ij} : Clinical Dementia Rating for subject (i) at time (j).

  • \beta_0 : Overall intercept (fixed effect).

  • \beta_1 : Fixed effect of Dementia Status.

  • \beta_2 : Fixed effect of Education Level.

  • \mu_{i} : Random intercept for each subject.

  • \epsilon_{ij} : Residual error term.

Data Cleaning:


#install.packages("readr")
library(readr)

oasis_data <- read_csv("oasis_longitudinal.csv")

# Remove duplicate rows
oasis_data <- oasis_data[!duplicated(oasis_data), ]

# Remove rows with missing values
oasis_data <- na.omit(oasis_data)

# View cleaned data
head(oasis_data)

1. Dementia Status as the only predictor


#install.packages("lme4")

library(lme4)
oasis_data$Group <- factor(oasis_data$Group)
oasis_data$SubjectID <- oasis_data$'Subject ID'

# Fit the linear mixed model with only Dementia Status as the predictor, reml
model <- lmer(CDR ~ Group + (1 | SubjectID), data = oasis_data)
summary(model)
Linear mixed model fit by REML ['lmerMod']
Formula: CDR ~ Group + (1 | SubjectID)
   Data: oasis_data

REML criterion at convergence: -141.5

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.2557 -0.4745 -0.0123 -0.0079  5.3656 

Random effects:
 Groups    Name        Variance Std.Dev.
 SubjectID (Intercept) 0.01641  0.1281  
 Residual              0.02604  0.1614  
Number of obs: 354, groups:  SubjectID, 142

Fixed effects:
                 Estimate Std. Error t value
(Intercept)       0.25057    0.04372   5.732
GroupDemented     0.42252    0.04914   8.597
GroupNondemented -0.24607    0.04778  -5.150

Correlation of Fixed Effects:
            (Intr) GrpDmn
GroupDemntd -0.890       
GropNndmntd -0.915  0.814

equation and interpretation of Dementia Status as the only predictor


the linear mixed equation is: \text{CDR}{ij} = 0.2506 + 0.4225 \cdot \text{Demented}{ij} - 0.2461 \cdot \text{Nondemented}{ij} + \mu_{i} + \epsilon_{ij}

where:

• If the group is “Demented”, the coefficient for Group Demented is used.

• If the group is “Nondemented”, the coefficient for Group Nondemented is used.

• If the group is the baseline (which is typically “Other” or the group not specifically named in the output), then no extra term is added for Group.


INTERPRETATION:

• The coefficient for Group Demented (0.4225) indicates that, on average, individuals in the “Demented” group have a higher CDR score by 0.4225 compared to the baseline group.

• The coefficient for Group Nondemented (-0.2461) indicates that, on average, individuals in the “Nondemented” group have a lower CDR score by 0.2461 compared to the baseline group.

• t-value for Demented = 8.597

• t-value for Nondemented = -5.150

• This shows a strong association between Dementia Status and CDR.

2.Age and Socio-Economic status as predictors


  • lmer function uses REML unless specified otherwise.
# Fit Linear Mixed Model with Age and Socio Economic status as the predictors
model <- lmer(CDR ~ Group + EDUC + (1 | SubjectID), data = oasis_data)
summary(model)
Linear mixed model fit by REML ['lmerMod']
Formula: CDR ~ Group + EDUC + (1 | SubjectID)
   Data: oasis_data

REML criterion at convergence: -134.6

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.2818 -0.4040 -0.0276  0.0290  5.3745 

Random effects:
 Groups    Name        Variance Std.Dev.
 SubjectID (Intercept) 0.01626  0.1275  
 Residual              0.02603  0.1614  
Number of obs: 354, groups:  SubjectID, 142

Fixed effects:
                  Estimate Std. Error t value
(Intercept)       0.149869   0.086623   1.730
GroupDemented     0.431773   0.049489   8.725
GroupNondemented -0.245532   0.047645  -5.153
EDUC              0.006606   0.004910   1.345

Correlation of Fixed Effects:
            (Intr) GrpDmn GrpNnd
GroupDemntd -0.564              
GropNndmntd -0.468  0.807       
EDUC        -0.864  0.139  0.009

equation and interpreation of Dementia Status and Education Level:


The model equation: \text{CDR}{ij} = 0.1499 + 0.4318 \cdot \text{Demented}{ij} - 0.2455 \cdot \text{Nondemented}{ij} + - 0.0066 \cdot \text{EDUC}{ij} + \mu_{i} + \epsilon_{ij}

INTERPRETATION

• The coefficient for Group Demented (0.4318) indicates that, on average, individuals in the “Demented” group have a higher CDR score by 0.4318 compared to the baseline group.

• The coefficient for Group Nondemented (-0.2455) indicates that, on average, individuals in the “Nondemented” group have a lower CDR score by 0.2455 compared to the baseline group.

• The coefficient for Education Level (0.0066), for each additional year of education, a subject’s CDR score increases by 0.00661 units on average.

• t-value for Demented = 8.725

• t-value for Nondemented = -5.153

• t-value for EDUC = 1.345

Conclusion


• Dementia Status (Group) is strongly associated with CDR.

• Education (EDUC) has a small, positive association with CDR, but this effect is not statistically significant (the t-value of 1.345 is below the typical threshold for significance).

References

  • Agostinelli, C., & Yohai, V. J. (2016). Composite robust estimators for linear mixed models. Journal of the American Statistical Association, 111(516), 1764-1774. https://doi.org/10.1080/01621459.2015.1115358

  • Bates, D. (2014). Fitting linear mixed-effects models using lme4. arXiv preprint arXiv:1406.5823.

  • Baayen, R. H., Davidson, D. J., & Bates, D. M. (2008). Mixed-effects modeling with crossed random effects for subjects and items. Journal of Memory and Language, 59(4), 390-412.

  • Barr, D. J., Levy, R., Scheepers, C., & Tily, H. J. (2013). Random effects structure for confirmatory hypothesis testing: Keep it maximal. Journal of Memory and Language, 68(3), 255-278.

  • Bruin, J. (2006). newtest: command to compute new test. UCLA: Statistical Consulting Group. https://stats.oarc.ucla.edu/stata/ado/analysis/.

References (continued)

  • Enders, C. K. (2010). Applied Missing Data Analysis. Guilford Press.

  • Gelman, A., & Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.

  • Starkweather, J. (2010). Linear mixed effects modeling using R. Unpublished Manuscript.